Shallow-Water Wave Problem

Problem Statement

An ocean wave propagates over a coastal shelf where the water depth is $h = 4.0\ \text{m}$ and the wavelength is $L = 150\ \text{m}$. Assume gravitational acceleration $g = 9.81\ \text{m/s}^{2}$.

Determine:

Whether the wave satisfies the shallow-water condition.
The wave speed.
The physical interpretation of the result.

Solution

Step 1: Shallow-Water Condition

$$ \frac{h}{L} \le 0.05 $$ $$ \frac{h}{L} = \frac{4.0}{150} = 0.0267 $$

Since $0.0267 < 0.05$, the wave is in the shallow-water regime.

Step 2: Wave Speed

For shallow-water waves:

$$ c = \sqrt{gh} $$ $$ c = \sqrt{(9.81)(4.0)} = \sqrt{39.24} $$ $$ \boxed{c \approx 6.26\ \text{m/s}} $$

Step 3: Physical Interpretation

Wave speed depends only on water depth.
Wavelength does not affect the speed.
Shallow-water waves are non-dispersive.
As the wave moves into shallower water, it slows and steepens.

Optional Check: General Formula